> > > Matheology § 230> > Cantor's theory of infinite sets, developed in the late 1800's, was a> decisive advance for mathematics, but it provoked raging controversies> and abounded in paradox. One of the first books by the distinguished> French mathematician Emile Borel (1871-1956) was his Lecons sur la> Théorie des Fonctions [Borel, 1950], originally published in 1898, and> subtitled Principes de la théorie des ensembles en vue des> applications à la théorie des fonctions.> This was one of the first books promoting Cantor's theory of sets> (ensembles), but Borel had serious reservations about certain aspects> of Cantor's theory, which Borel kept adding to later editions of his> book as new appendices. The final version of Borel's book, which was> published by Gauthier-Villars in 1950, has been kept in print by> Gabay. That's the one that I have, and this book is a treasure trove> of interesting mathematical, philosophical and historical material.> One of Cantor's crucial ideas is the distinction between the> denumerable or countable infinite sets, such as the positive integers> or the rational numbers, and the much larger nondenumerable or> uncountable infinite sets, such as the real numbers or the points in> the plane or in space. Borel had constructivist leanings, and as we> shall see he felt comfortable with denumerable sets, but very> uncomfortable with nondenumerable ones.

Since denumerable sets are infinite sets, and borel sets allow denumberable union or intersections of intervals which are themselves nondenumerable sets, Borel could not have been too nervous about non-denumerability.